# Local Linearizations of Rational Matrices with Application to Rational Approximations of Nonlinear Eigenvalue Problems

## TL;DR

This paper introduces a comprehensive framework for local linearizations of rational matrices, enabling structure-preserving approximations of zeros, poles, and eigenvalues, with applications to nonlinear eigenvalue problems.

## Contribution

It provides a unified definition of local linearizations that encompasses previous approaches and rigorously explains their properties, especially in the context of nonlinear eigenvalue problems.

## Key findings

- New definition of local linearizations for rational matrices.
- Unified framework explaining properties of existing pencils.
- Application to rational approximation of nonlinear eigenvalue problems.

## Abstract

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows us to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. Moreover, such definition includes, as particular cases, other definitions that have been used previously in the literature. In this way, this new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2019 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.